During the nuclear explosion, one of the products is ${ }^{90}\mathrm{Sr}$  with half life of 6.93 years , if  $1\mathrm{\mu g}$of  ${ }^{90}\mathrm{Sr}$ was absorbed in the bones of a newly born baby in place of Ca, how much time, in years, is required to reduce it by 90% if it is not lost metabolically ____.

To determine how long it takes for the amount of 90^Sr to decrease by 90%, we need to understand the concept of half-life and apply it to radioactive decay.

Key Concepts

• Half-Life: The time required for half of the radioactive atoms in a sample to decay. For 90^Sr, this is 6.93 years.
• Radioactive Decay: A random process where unstable atomic nuclei lose energy by emitting radiation. The decay rate is constant and independent of the amount of substance present.
• First-Order Kinetics: Radioactive decay follows first-order kinetics, meaning the rate of decay is proportional to the amount of substance present.

Calculations

We can use the following formula to calculate the time required for a substance to decay to a certain level:

t = (2.303 / k) × log₁₀(C₀ / C)

Where:

• t: Time required for the decay (in years)
• k: Decay constant (in years^-1)
• C₀: Initial amount of 90^Sr (1 μg)
• C: Final amount of 90^Sr after decay (10% of initial amount, i.e., 0.1 μg)

Step 1: Calculate the Decay Constant (k)

The decay constant is related to the half-life (t₁/₂) by the equation:

k = 0.693 / t₁/₂

Substituting the given half-life of 6.93 years:

k = 0.693 / 6.93 ≈ 0.1 years^-1

Step 2: Apply the Formula

Now, we can substitute the values into the decay formula:

t = (2.303 / 0.1) × log₁₀(1 / 0.1)

Calculating the logarithm:

log₁₀(1 / 0.1) = log₁₀(10) = 1

Therefore:

t = (2.303 / 0.1) × 1 = 23.03 years

It will take approximately 23 years for the amount of 90^Sr in the bones to decrease by 90% due to its radioactive decay, assuming it is not lost metabolically.